class_docstr: Abstract Mixin class for acquisition functions using a cached Cholesky.
  
  Specifically, this is for acquisition functions that require sampling from
  the posterior P(f(X_baseline, X) | D). The Cholesky of the posterior
  covariance over f(X_baseline) is cached.
  
  :meta private:
docstr: Get posterior samples at the `q_in` new points from the joint posterior.
  
  Args:
      posterior: The joint posterior is over (X_baseline, X).
      q_in: The number of new points in the posterior. See `_set_sampler` for
          more information.
  
  Returns:
      A `sample_shape x batch_shape x q x m`-dim tensor of posterior
          samples at the new points.
docstr_tok: ["Get","posterior","samples","at","the","`","q_in","`","new","points","from","the","joint","posterior",".","Args",":","posterior",":","The","joint","posterior","is","over","(","X_baseline",",","X",")",".","q_in",":","The","number","of","new","points","in","the","posterior",".","See","`","_set_sampler","`","for","more","information",".","Returns",":","A","`","sample_shape","x","batch_shape","x","q","x","m","`","-dim","tensor","of","posterior","samples","at","the","new","points","."]
code: def _get_f_X_samples(
      self, posterior: GPyTorchPosterior, q_in: int
  ) -> Tensor:
      r"""Get posterior samples at the `q_in` new points from the joint posterior.
  
      Args:
          posterior: The joint posterior is over (X_baseline, X).
          q_in: The number of new points in the posterior. See `_set_sampler` for
              more information.
  
      Returns:
          A `sample_shape x batch_shape x q x m`-dim tensor of posterior
              samples at the new points.
      """
      # Technically we should make sure that we add a consistent nugget to the
      # cached covariance (and box decompositions) and the new block.
      # But recomputing box decompositions every time the jitter changes would
      # be quite slow.
      if self._cache_root and hasattr(self, "_baseline_L"):
          try:
              return sample_cached_cholesky(
                  posterior=posterior,
                  baseline_L=self._baseline_L,
                  q=q_in,
                  base_samples=self.sampler.base_samples,
                  sample_shape=self.sampler.sample_shape,
              )
          except (NanError, NotPSDError):
              warnings.warn(
                  "Low-rank cholesky updates failed due NaNs or due to an "
                  "ill-conditioned covariance matrix. "
                  "Falling back to standard sampling.",
                  BotorchWarning,
              )
  
      # TODO: improve efficiency for multi-task models
      samples = self.get_posterior_samples(posterior)
      if isinstance(self.model, HigherOrderGP):
          # Select the correct q-batch dimension for HOGP.
          q_dim = -self.model._num_dimensions
          q_idcs = (
              torch.arange(-q_in, 0, device=samples.device)
              + samples.shape[q_dim]
          )
          return samples.index_select(q_dim, q_idcs)
      else:
          return samples[..., -q_in:, :]
  
imports: from __future__ import annotations
  import warnings
  from typing import Optional
  import torch
  from botorch.acquisition.acquisition import MCSamplerMixin
  from botorch.exceptions.warnings import BotorchWarning
  from botorch.models.gpytorch import GPyTorchModel
  from botorch.models.higher_order_gp import HigherOrderGP
  from botorch.models.model import Model
  from botorch.models.model_list_gp_regression import ModelListGP
  from botorch.models.multitask import KroneckerMultiTaskGP, MultiTaskGP
  from botorch.posteriors.gpytorch import GPyTorchPosterior
  from botorch.posteriors.posterior import Posterior
  from botorch.sampling.base import MCSampler
  from botorch.utils.low_rank import extract_batch_covar, sample_cached_cholesky
  from gpytorch.distributions.multitask_multivariate_normal import MultitaskMultivariateNormal
  from linear_operator.utils.errors import NanError, NotPSDError
  from torch import Tensor
label_desc: Use image node_id 3 for calling the CachedCholeskyMCSamplerMixin obj's underlying member method code with example usage: obj._get_f_X_samples(posterior, q_in) and returns: samples, samples, sample_cached_cholesky